1. Information criteria What function fits best? The more free parameters a model has the higher will be R 2. The more parsimonious a model is the lesser is the bias towards type I errors. We have to find a compromis between goodness of fit and bias! Model parameters few many Bias Explained variance The optimal number of model parameters

2. The Akaike criterion of model choice k: number of model parameters +1 L: maximum likelihood estimate of the model If the parameter errors are normal and independent we get n: number data points RSS: residual sums of squares If we fit using  2 : If we fit using R 2 : At small sample size we should use the following correction The preferred model is the one with the lowest AIC.

3. We get the surprising result that the seemingly worst fitting model appears to be the preferred one. A single outlier makes the difference. The single high residual makes the exponential fitting worse

4. Significant difference in model fit Approximately  AIC is statisticaly significant in favor of the model with thesmaller AIC at the 5% error benchmark if |  AIC| > 2. The last model is not significantly (5% level) different from the second model. AIC model selection serves to find the best descriptor of observed structure. It is a hypothesis generating method. It does not test for significance Model selection using significance levels is a hypothesis testing method. Significance levels and AIC must not be used together.

5. Literature